Lesson 4 of 5
Rewrite equivalent expressions while keeping every original domain restriction.
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Big ideas
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An IdentityAn equation that works for every allowed input.Exactly: An equality true for every value in the common domain of its two sides.Mix-up alert: Checking one angle can disprove an identity but cannot prove it.See this in Key words holds for every input in the common domain of its two sides. A few matching numerical values can suggest a pattern but do not prove it.
Use the IdentityAn equation that works for every allowed input.Exactly: An equality true for every value in the common domain of its two sides.Mix-up alert: Checking one angle can disprove an identity but cannot prove it.See this in Key words . Dividing by gives , where is nonzero; dividing by gives , where is nonzero.
Start with the more complicated ExpressionA mathematical phrase made from numbers, variables and operations.Exactly: A combination of numbers, variables and operations representing a value when defined.Mix-up alert: An expression is not an equation; it has no equals sign.See this in Key words and replace ratios, factor, or use common denominators. Cancel common factors, never individual terms of a sum.
A simplified ExpressionA mathematical phrase made from numbers, variables and operations.Exactly: A combination of numbers, variables and operations representing a value when defined.Mix-up alert: An expression is not an equation; it has no equals sign.See this in Key words may be defined at more inputs than the original. Keep the original exclusions when claiming equivalence, such as not equal to 0 after canceling a factor .
Step-through
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Simplify , preserving its domain.
Whole picture, 4 steps
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Worked example
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Worked example
Do this: Follow each step on paper before reading the result.
Mix-up alert
Do this: Explain the wrong idea and the repair.
Wrong idea: matching a graph or checking three angles proves an identity. Repair: numerical checks can miss excluded inputs and other failures. Give a valid algebraic chain on the common domain.
Tip
Do this: Use paper. State the definition before you calculate.
Explain why checking an identity at one angle is not a proof for every allowed angle.
Worked example
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Simplify .
Watch
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State the basic Pythagorean identity.
for every real .
Rewrite .
.
Rewrite .
, where is nonzero.
Rewrite .
, where is nonzero.
What is ?
, where is nonzero.
What is ?
, where is nonzero.
Can you cancel the term from ?
No. It is a term in a sum, not a factor of the entire numerator.
After cancellation, which domain should you report?
Keep all original denominator exclusions.
True or false
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Self-test
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6 questions. One at a time.
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Self-test
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6 questions. One at a time.
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Write it
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Write it
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Match
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Connect the ideas
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